Which of the following describes the appearance of the graph at a zero whose multiplicity is four? Enter the number of the correct option.
Option #1: The graph crosses the x-axis.
Option #2: The graph touches the x-axis and turns around.
Option #3: There is not enough information provided.
Option #2: The graph touches the x-axis and turns around.
Option #2: The graph touches the x-axis and turns around.
To determine the appearance of the graph at a zero whose multiplicity is four, we need to understand the concept of multiplicity and its impact on the graph.
In algebra, the term "multiplicity" refers to the number of times a particular root or zero appears in a polynomial equation. Multiplicity affects the behavior of the graph at that specific zero.
In this case, since the multiplicity is four, we can determine the appearance of the graph by considering the exponent of the factor corresponding to the zero. Let's break down each option to see which one is the most suitable:
Option #1: The graph crosses the x-axis.
If the graph crosses the x-axis at a zero with a multiplicity of four, it means the exponent of the factor corresponding to the zero is odd. In this case, the graph would cross the x-axis.
Option #2: The graph touches the x-axis and turns around.
If the graph touches the x-axis and turns around at a zero with a multiplicity of four, it means the exponent of the factor corresponding to the zero is even. In this case, the graph would touch the x-axis and change direction.
Option #3: There is not enough information provided.
Based on the information given, we do have enough information to determine the correct option.
Given that the multiplicity of the zero is four, we can see that the exponent is an even number. Therefore, the correct option is "Option #2: The graph touches the x-axis and turns around."
So, the correct answer is Option #2.